Energy-conserving time propagation for a structure-preserving particle-in-cell Vlasov–Maxwell solver

This paper discusses energy-conserving time-discretizations for finite element particle-in-cell discretizations of the Vlasov–Maxwell system. A geometric spatially discrete system can be obtained using a standard particle-in-cell discretization of the particle distribution and compatible finite element spaces for the fields to discretize the Poisson bracket of the Vlasov–Maxwell model (see Kraus et al. (2017) [1]). In this paper, we derive energy-conserving time-discretizations based on the discrete gradient method applied to an antisymmetric splitting of the Poisson matrix. Firstly, we propose a semi-implicit method based on a splitting that yields constant Poisson matrices in each substep. Moreover, we devise an alternative discrete gradient that yields a time discretization that can additionally conserve Gauss' law. Finally, we explain how substepping for fast species dynamics can be incorporated.